The rules for a book report say that the report should have 300 words with an absolute value deviation of at most 20 words. Write and solve an absolute value inequality that represents the acceptable number of words.Please help me!
|w-300| <= 20
So, what are two numbers that are 20 away from 300?
280 and 320
To write an absolute value inequality that represents the acceptable number of words for the book report, we can use the formula:
|number of words - target value| <= deviation
In this case, the target value is 300 words, and the maximum deviation allowed is 20 words. Therefore, the absolute value inequality can be written as:
|number of words - 300| <= 20
To solve this inequality, we can break it down into two separate inequalities:
1) number of words - 300 <= 20
2) -(number of words - 300) <= 20
Solving the first inequality:
number of words - 300 <= 20
Adding 300 to both sides:
number of words <= 320
Solving the second inequality:
-(number of words - 300) <= 20
Expanding the absolute value:
- number of words + 300 <= 20
Subtracting 300 from both sides:
- number of words <= -280
Dividing both sides by -1 (since we multiply or divide by a negative number, the inequality sign flips):
number of words >= 280
Therefore, the solution to the absolute value inequality is given by:
280 <= number of words <= 320
To write an absolute value inequality that represents the acceptable number of words, we need to consider the allowed deviation around the desired 300-word count.
Let's denote x as the number of words in the book report.
The absolute value deviation from the desired 300 words can be represented as |x - 300|.
The absolute value inequality for the acceptable number of words is:
| x - 300 | ≤ 20
This inequality states that the absolute value of the difference between the number of words in the book report and 300 should be less than or equal to 20.
To solve this inequality, we can consider two cases:
1. x - 300 ≤ 20:
Solving for x, we add 300 to both sides:
x ≤ 320
2. -(x - 300) ≤ 20 (using the negative case of the absolute value):
Expanding the expression, we get:
-x + 300 ≤ 20
Subtracting 300 from both sides, we have:
-x ≤ -280
Multiplying both sides by -1 (and flipping the inequality sign as a result), we get:
x ≥ 280
Thus, the solution to the absolute value inequality |x - 300| ≤ 20 is:
x ≤ 320 or x ≥ 280.
This means that the book report can have a word count between 280 and 320 words, inclusive, to meet the requirements.